272 research outputs found
Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires
In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large
Lagrangian and Eulerian Descriptions in Solid Mechanics and Their Numerical Solutions in hpk Framework
In this thesis mathematical models for a deforming solid medium are derived using conservation laws in Lagrangian as well as Eulerian descriptions. First, most general forms of the mathematical models permitting compressibility of the matter are considered which are then specialized for incompressible medium. Development of constitutive equations central to the validity of the mathematical models is considered Numerical solution of these mathematical models are obtained using finite element method based on h,p,k mathematical and computational framework in which the integral forms are variationally consistent and hence the resulting computational processes are unconditionally stable. Details of the constitutive equations in both Lagrangian and Eulerian descriptions are presented. A variety of model problems are chosen for numerical studies. The wave propagation model problems are considered for numerical studies to investiage (i) Behaviors and limitations of constitutive models in both descriptions (ii) Overall benefits and drawbacks of Lagrangian and Eulerian descriptions
Recent mathematical developments in the Skyrme model
In this review we present a pedagogical introduction to recent, more
mathematical developments in the Skyrme model. Our aim is to render these
advances accessible to mainstream nuclear and particle physicists. We start
with the static sector and elaborate on geometrical aspects of the definition
of the model. Then we review the instanton method which yields an analytical
approximation to the minimum energy configuration in any sector of fixed baryon
number, as well as an approximation to the surfaces which join together all the
low energy critical points. We present some explicit results for B=2. We then
describe the work done on the multibaryon minima using rational maps, on the
topology of the configuration space and the possible implications of Morse
theory. Next we turn to recent work on the dynamics of Skyrmions. We focus
exclusively on the low energy interaction, specifically the gradient flow
method put forward by Manton. We illustrate the method with some expository toy
models. We end this review with a presentation of our own work on the
semi-classical quantization of nucleon states and low energy nucleon-nucleon
scattering.Comment: 129 pages, about 30 figures, original manuscript of published Physics
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